Involves finding the intervals where a function f is increasing and decreasing. Given the function f(x)={(x+7,if x lt-3,,),(|x+1|,if -3 leq x<1,,),(5-2x,if x ge 1,,):}

omvamen71 2022-09-28 Answered
Increasing and decreasing piecewise function on an interval
I'm working on a problem that involves finding the intervals where a function f is increasing and decreasing. Given the function f ( x ) = { x + 7  \text{if } x\lt -3 | x + 1 |  \text{if } -3\le x <1 5 2 x  \text{if } x\ge 1
I worked out that f is increasing on ( , 3 ) and [ 1 , 1 ), and f is decreasing on [ 3 , 1 ] , [ 1 , ).
However, the solution in the book claims that f is increasing on [-1,1], rather than [-1,1) like I worked out. I am having trouble understanding why the book claims that this is the case.
I was under the impression that f must be differentiable on the interior of an interval I and continuous on all of I in order to make any statements about increasing/decreasing behavior on the closed interval I. I'm not sure if I am overlooking something, but it seems that f is not continuous on [-1,1]. I would appreciate any clarification.
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Answers (2)

Alvin Preston
Answered 2022-09-29 Author has 9 answers
Step 1
It is true that if you have a differentiable function on an interval, then it is increasing if and only if its derivative is non-negative. However, increasing functions need not be differentiable according to their definition:
A function f : R R is increasing on a collection S if and only if:
For any x , y S such that x y:
f ( x ) f ( y )
Step 2
Note that this definition is incompatible with the one that lulu proposed in a comment, but I believe this is the typical definition.
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Deanna Gregory
Answered 2022-09-30 Author has 3 answers
Explanation:
You have found that f is increasing on [ 1 , 1 )
Now lim x 1 f ( x ) = | 1 + 1 | = 2 < 3 = f ( 1 ).
Thus f is increasing on [ 1 , 1 ]
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